Find the equation of the exponential function represented by the table below:
x
x
y
y
0
0
0.01
0.01
1
1
0.005
0.005
2
2
0.0025
0.0025
3
3
0.00125
0.00125
There are different ways to approach this problem, but one common method is to use the general form of an exponential function:
y = a ⋅ b^x
where a is the initial value (when x=0), b is the base of the exponential (a constant greater than 0), and x is the independent variable (usually representing time or another variable that changes continuously).
To find the value of a, we can see from the table that y=0 when x=0, so a=0. To find the value of b, we can notice that each y-value is half of the previous y-value, meaning that the ratio between consecutive y-values is the same:
1/0.005 = 200
0.005/0.0025 = 2
0.0025/0.00125 = 2
This common ratio is equal to b^1 (since it is the ratio between consecutive outputs for a one-unit increase in the input). Therefore:
b^1 = 2
b = 2^(1/1) = 2
Substituting a=0 and b=2 into the general form of an exponential function, we get:
y = 0 ⋅ 2^x = 0
This means that the exponential function represented by the table is y = 0 for all values of x. Alternatively, we can notice that the values in the table are decreasing exponentially as x increases, which means that the function could be expressed as the inverse of an exponential function, namely:
y = 1/(2^x)
To find the equation of the exponential function represented by the table, we need to determine the exponential growth rate.
Let's start by looking at the ratios between consecutive y-values. Notice that each ratio is the same, which indicates exponential growth.
y(0.01)/y(0) = 0.01/0 = undefined (cannot divide by zero)
y(1)/y(0.01) = 1/0.01 = 100
y(0.005)/y(1) = 0.005/1 = 0.005
y(0.0025)/y(0.005) = 0.0025/0.005 = 0.5
y(0.00125)/y(0.0025) = 0.00125/0.0025 = 0.5
Since the ratios are the same, we can conclude that the growth rate is constant.
To find the growth rate, we need to calculate the common ratio (r). Taking any two consecutive y-values, we have:
r = y(1)/y(0.01) = 100
Now, let's find the initial value (a) of the exponential function. We can pick any two corresponding x- and y-values. Let's choose x = 0 and y = 0.
Using the exponential growth formula y = a * r^x, we can substitute the values to solve for a:
0 = a * 100^0
0 = a * 1
a = 0
Therefore, the equation of the exponential function represented by the table is:
y = 0 * 100^x
y = 0
So, the equation simplifies to:
y = 0